Exercise 8 | Wiskunde op Tilburg University

Determine all $a$ such that the following equation has exactly two solutions: $\dfrac{x}{5}+\dfrac{a^2}{x}=2$ .

Antwoord 1 correct

Correct

Antwoord 2 optie

$-\sqrt{5}$

Antwoord 2 correct

Fout

Antwoord 3 optie

For all $a$ .

Antwoord 3 correct

Fout

Antwoord 4 optie

For no $a$ .

Antwoord 4 correct

Fout

Antwoord 1 optie

The correct answer is not among the other options.

Antwoord 1 feedback

Correct: Note that the equation is not defined for $x=0$ . Hence, for no $a$ is $x=0$ a solution.

If $a=0$ : $\dfrac{x}{5}=2$ has one solution.

For $a\neq 0$ we have $\dfrac{x}{5}+\dfrac{a^2}{x}=2$ .

Rewriting by multiplying by $x$ gives $\frac{1}{5}x^2-2x+a^2=0$ .

$D=4-\frac{4}{5}a^2$ .

$D=0$ for $a=-\sqrt{5}$ or $a=\sqrt{5}$ .

$D>0$ for $-\sqrt{5}<a<\sqrt{5}$ .

Hence, two solutions for $-\sqrt{5}<a<0$ or $0<a<\sqrt{5}$ .
Go on.

Antwoord 2 feedback

Wrong: What is the equation for $a=0$ ?

Try again.

Antwoord 3 feedback

Wrong: Try again.

Antwoord 4 feedback

Wrong: Try again.